Periodic solutions for SDEs through upper and lower solutions

Chunyan Ji; Yang Xue; Yong Li

doi:10.3934/dcdsb.2020122

Article Contents

2020, Volume 25, Issue 12: 4737-4754. Doi: 10.3934/dcdsb.2020122

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Periodic solutions for SDEs through upper and lower solutions

1.
School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
2.
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
3.
School of Mathematics, Jilin University, Changchun 130012, China
4.
State Key Laboratory of Automotive Simulation and Control, Jilin University, 130025, China

^* Corresponding author: Yong Li
^* Corresponding author: Yong Li

Received: October 31, 2019

Revised: December 31, 2019

Early access: March 2020

Published: December 2020

The first author is supported NSFC grant 11601043, China Postdoctoral Science Foundation (Grant No. 2016M590243 and 2019T120226). The second author was supported by NSFC grant 11201173. The third author was supported by National Basic Research Program of China (Grant No. 2013CB834100) and NSFC grants 11171132 and 11571065

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Abstract

We study a kind of better recurrence than Kolmogorov's one: periodicity recurrence, which corresponds periodic solutions in distribution for stochastic differential equations. On the basis of technique of upper and lower solutions and comparison principle, we obtain the existence of periodic solutions in distribution for stochastic differential equations (SDEs). Hence this provides an effective method how to study the periodicity of stochastic systems by analyzing deterministic ones. We also illustrate our results.

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Mathematics Subject Classification: Primary: 60H10, 34C27; Secondary: 37B25.

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